In this paper,we investigate duplex mode selection and transceiver design for cell-free massive multiple-input multiple-output(MIMO)with network-assisted full duplexing(NAFD),where the remote antenna units(RAUs)simultaneously serve both uplink and downlink users on the same time-frequency resource,and the joint transmission and reception are done at the central processing unit.We consider that each antenna can operate in three modes,i.e.,uplink reception,downlink transmission and sleep.In particular,we model the problem as a mixed-integer optimization problem to maximize the aggregated spec-tral efficiency(SE)of downlinks and uplinks,where the quality-of-service constraints and power budget constraints are considered.Since the uplinks and downlinks are highly coupled,to address the design prob-lem,we resolve the combinatorial problem by a series of nonconvex-convex approximate methods,such as equivalent formulations,iterative success convex approximations,and binary relaxations.In particular,a two-stage strategy is proposed to solve the optimization problem.In the first stage,we fix the mode selection vectors and then obtain the optimal transceivers with such a fairness strategy.In the second stage,with the transceiver parameters obtained from the first stage,we further optimize the duplex mode of each antenna.The algorithms of the two stages will alternately run in several loops until convergence.Numerical results indicate that the proposed solution can obtain SE performance that is close to the optimal exhaustive search solution and yields a higher SE gain compared with the traditional fixed-mode duplex scheme.